Rethinking Pythagoras - Appendix: Discovering Pythagoras

Here we give a discovery activity that we have used, in its entirety, with undergraduate students studying non-Euclidean geometry. We use freeware Geometry Playground [1] for these investigations; simply select the main link on the Geometry Playground page to run it. Geometry Playground allows straightedge and compass construction in Euclidean and non-Euclidean geometries such as spherical and hyperbolic. We let \(r\) represent the radius of a circle as measured in each geometry, while \(C\) and \(A\) represent the circumference and area of that circle, again as measured within that geometry.

This link will open Geometry Playground in Euclidean geometry with a pre-constructed right triangle having vertices \(A\), \(B\), and \(C\), the last corresponding to the right angle, and a constant term of \(\pi\) that can be used in forming sums, differences, products, and ratios. Construct three circles, one with center \(A\) and containing point \(B\), another with center \(C\) containing \(A\), and the third with center \(B\) containing point \(C\). Measure the areas of these circles.

Denote the length of the side with vertices \(B\) and \(C\) by \(a\), the side with vertices \(A\) and \(C\) by \(b\), and the third, the hypotenuse, by \(c\). Write the areas of the circles in terms of \(a\), \(b\), and \(c\). How do you know that the expressions you wrote down are correct?

Find a relationship between the areas of the circles. Make sure that the relationship you find remains after moving points in the triangle about. Write the relationship in terms of \(a\), \(b\), and \(c\).Hint: Sum the areas of the two smaller circles (you can do this by selecting Measure → Sum) and compare the sum to the area of the larger circle, perhaps by finding the ratio or the difference.

Select Spherical geometry and choose the plane model by selecting Display → Model → Plane Model. Scale up as far as possible, so that you are looking at a very, very tiny part of the unit sphere. Construct a circle, and measure its radius, circumference, and area. Do \(C=2\pi r\) and \(A=\pi r^2\)? (Note that you can study the ratios of measures dynamically by selecting Measure → Ratio.) What happens as you scale down and look at larger and larger circles?

Note that when \(r\) is very small, we can estimate circumference and area using familiar formulas. What are they?

Switch back to the standard (sphere) model of spherical geometry. When \(r=\pi/2\), the circle is a great circle; what are circumference and area? How about when \(r=\pi\)?

Use these known facts to conjecture an expression for \(C\) with respect to \(r\). Test your conjecture for several values of \(r\) using Geometry Playground.Hint: Since circumference grows and then shrinks, a trigonometric function might be useful here.

Note that in Euclidean geometry, \(dA/dr=C\). Explain why. A picture may be helpful.

Explain why your answer to part (d) remains true in spherical geometry (even though the expression for \(C\) is different).

Use the results of (c)-(e) to find an expression for \(A\) with respect to \(r\). Test your answer for several values of \(r\) using Geometry Playground.

This link will open Geometry Playground in spherical geometry with a pre-constructed right triangle having vertices \(A\). \(B\), and \(C\), the last corresponding to the right angle, and a constant term of \(2\pi\) that can be used in forming sums, differences, products, and ratios. Again construct three circles with radii the sides of the triangle, and measure the areas of those circles. For each circle \(S_i\), calculate \([2\pi-A(S_i)]/(2\pi)\), hiding all intermediate calculations.

If we denote by \(S_0\) and \(S_1\) the circles constructed with radii the legs of the right triangle, and \(S_2\) the circle whose radius is the hypotenuse, note that \([2\pi-A(S_0)]/(2\pi)\cdot [2\pi-A(S_1)]/(2\pi)=[2\pi-A(S_2)]/(2\pi)\), that is, try it.
(Note: based on your results for part (f), you should be able to find a simple expression for \([2\pi-A(S_i)]/(2\pi)\) in terms of the length of the radius \(r_i\) of the circle \(S_i\).) Find an equivalent expression in terms of the lengths of the sides (\(a\), \(b\)) and hypotenuse (\(c\)) of the triangle. This is the "Spherical Pythagorean Theorem."

Find the Taylor series for the trigonometric expressions in (h). Verify that by ignoring the terms of degree larger than 2 (for example, when \(a\), \(b\), and \(c\) are small), we obtain the standard Pythagorean Theorem.

In Geometry Playground, spherical length measurements are all with respect to the radius of the sphere itself. That is, the sphere's radius is assumed to be 1 unit. Hence, if the radius of the sphere is 6371 kilometers (such as for the sphere we live on), then all measurements in kilometers should be divided by 6371 to obtain their equivalent in terms of radii length, and measurements in terms of sphere radius length should be multiplied by 6371 to obtain their kilometer equivalents. Find the length of the hypotenuse of a triangle on the surface of the earth that has sides of length 5000 km and 8000 km. How about 5 km and 8 km? How about 5 m and 8 m? At what point can we ignore the fact that we live on a sphere when making calculations?

Select Hyperbolic geometry and choose the model by selecting Display → Model → Minkowski-Weierstrass Model (also known as the hyperboloid model). Scale up as far as possible, so that you are looking at a very, very tiny part of the hyperbolic plane. Construct a circle, and measure its radius, circumference, and area. Do \(C=2\pi r\) and \(a=\pi r^2\)? What happens as you scale down and look at larger and larger circles?

Explain why the relationship in 2(d) still holds in hyperbolic geometry.

We note that the formula for circumference in hyperbolic geometry is \(C(r)=2\pi\sinh r\). Use this and your answer to part (a) to find an expression for the area of a circle in hyperbolic geometry in terms of the length of the radius. Note the similarity to your answer for 2(f). Test your answer for several values of \(r\) using Geometry Playground.

This link will open Geometry Playground in the Minkowski-Weierstrass model of hyperbolic geometry with a pre-constructed right triangle having vertices \(A\), \(B\), and \(C\), the last corresponding to the right angle, and a constant term of \(2\pi\) that can be used in forming sums, differences, products, and ratios. Again construct three circles with radii the sides of the triangle, and measure the areas of those circles. For each circle \(S_i\), calculate \([A(S_i)+2\pi]/(2\pi)\), hiding all intermediate calculations. Note: based on your results for part (b), you should be able to find a simple expression for \([A(S_i)+2\pi]/(2\pi)\) in terms of the length of the radius \(r_i\) of the circle \(S_i\).

If we denote by \(S_0\) and \(S_1\) the circles constructed with radii the legs of the right triangle, and \(S_2\) the circle whose radius is the hypotenuse, note that \([A(S_0)+2\pi]/(2\pi)\cdot [A(S_1)+2\pi]/(2\pi)=[A(S_2)+2\pi]/(2\pi)\), that is, try it. Find an equivalent expression in terms of the lengths of the sides (\(a\), \(b\)) and hypotenuse (\(c\)) of the triangle. This is the "Hyperbolic Pythagorean Theorem."

Find the Taylor series for the hyperbolic trigonometric expressions in (d). Verify that by ignoring the terms of degree larger than 2 (for example, when \(a\), \(b\), and \(c\) are small), we obtain the standard Pythagorean Theorem.